2025-02-25
We have previously shown that the value of an option is the risk-neutral expectation of its discounted payoff
In the binomial model this was seen in the single-period model
\[ C_{0} = e^{-rh}\left[p^{\ast} C_{u} + (1 - p^{\ast})C_{d}\right] \]
The term inside the brackets is the binomial expected value under the risk-neutral distribution
This suggests a path forward for models beyond the binomial model
We can obtain an estimate of the risk-neutral expected value by computing the average of a large number of discounted payoffs
This is a LLN argument based on frequentist reasoning
Consider an European call option that pays \(C_{T}\) at expiry.
First, we simulate the risk-neutral process of the state variables from their observed values today
Second, apply the payoff function to the terminal values simulated along each path and obtain \(C_{T,j}\) for path \(j\)
Then discount this payoff (assuming a constant risk-free rate):
\[ C_{0,j} = e^{-rT} C_{T,j} \]
\[ \hat{C}_{0} = \frac{1}{M} \sum\limits_{j=1}^{M} C_{0,j} \]
This is basically a proposal to evaluate the expectation statistically using Monte Carlo integration
There will thus be sampling error introduced in the algorithm and we can estimate it via the standard error of the simulation
\[ SE(\hat{C}_{0}) = \frac{SD(C_{0,j})}{\sqrt{M}} \]
The standard deviation is given by:
\[ \mathrm{SD}\left(C_{0,j}\right) = \left(\frac{1}{M-1} \sum_{j=1}^M\left(C_{0,j} - \hat{C}_0\right)^2\right)^{\frac{1}{2}} \]